A parity for 2-colourable links
Abstract
We introduce the 2-colour parity. It is a theory of parity for a large class of virtual links, defined using the interaction between orientations of the link components and a certain type of colouring. The 2-colour parity is an extension of the Gaussian parity, to which it reduces on virtual knots. We show that the 2-colour parity descends to a parity on free links. We compare the 2-colour parity to other parity theories of virtual links, focusing on a theory due to Im and Park. The 2-colour parity yields a strictly stronger invariant than the Im-Park parity. We introduce an invariant, the 2-colour writhe, that takes the form of a string of integers. The 2-colour writhe is a concordance invariant, and so obstructs sliceness. It is also an obstruction to amphichirality and chequerboard colourability within a concordance class.
Cite
@article{arxiv.1901.07406,
title = {A parity for 2-colourable links},
author = {William Rushworth},
journal= {arXiv preprint arXiv:1901.07406},
year = {2022}
}
Comments
36 pages, 15 figures. Comments welcome. Section 4.2 revised. This version to appear in the Osaka Journal of Mathematics